Day 5 ­ Nine Point Circle and Euler Line

1) Finding the orthocenter, incenter, and circumcenter
2) Construct the Nine Point Circle and Euler Line
3) Compare and explore with various types of triangles

1) Review with the class all the constructions performed since the start of the unit, medians, altitudes, perpendicular bisectors, angle bisectors and the centroid

2) Find the orthocenter, incenter, and circumcenter

Orthocenter-the point where the altitudes meet
Circumcenter-the point where the perpendicular bisectors meet
Incenter-the point where the angle bisectors meet

3) Have the students construct the following points and label them

 Description of Points

 Label of Points
 midpoints of the three sides (3)

 E, F, G
 circumcenter (1)

 centroid (1)

 intersection points of altitudes with the sides (3)

 N, P, Q
 orthocenter (1)

 midpoints of segments from orthocenter to each vertex (3)

 U, V, W
 incenter (1)

 midpoint of segment joining circumcenter and orthocenter (1)


Click here for Step by Step Instructions that can be used in the classroom

4) Once all points have been located have the students construct a segment from the circumcenter to the orthocenterthis segment is the Euler Line

5) Then construct a circle with the center at point Z and the radius goes from point Z to the midpoint of a side of the triangleforming our Nine point circle

6) Have the students explore these constructions and what happens when you have different size and types of triangles

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